2016/09/06 by Loeb, Sarah J., West, Douglas B.
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1609.01612
The separation dimension of a graph G, written π(G), is the minimum number of linear orderings of V(G) such that every two nonincident edges are "separated" in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the fractional separation dimension πf(G), which is the minimum of a/b such that some a linear orderings (repetition allowed) separate every two nonincident edges at least b times. In contrast to separation dimension, fractional separation dimension is bounded: always πf(G)≤ 3, with equality if and only if G contains K4. There is no stronger bound even for bipartite graphs, since πf(Km,m)=πf(Km+1,m)=(3m)/(m+1). We also compute πf(G) for cycles and some complete tripartite graphs. We show that πf(G)